Thermostat model with one sensor

This section studies the existence and uniqueness results to the DFTM with three-point BCs (2). First, we introduce some notations that are used in this paper. Let (mathcal{B}) be a Banach space with norm (Vert u Vert =max vert u(kappa ) vert ) for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}). Now, we state and prove an important theorem that deals with a linear variant of the solution of DFTM with three-point BCs (2) and we give a representation of the solution.

Theorem 3.1

Let realvalued function (mathcal{F}) be defined on (mathbb{N}_{vartheta -2}^{vartheta +ell +1}). Then, for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}) the following DFTM

$$begin{aligned} textstylebegin{cases} – {}^{C}Delta ^{vartheta }u(kappa )=mathcal{F}(kappa +vartheta -1),quad kappa in {mathbb{N}_{0}^{ell +1}}, \ Delta u(vartheta -2)=0, qquad delta {}^{C}Delta ^{vartheta -1} u( vartheta +ell )+gamma u(eta )=0, end{cases}displaystyle end{aligned}$$

(4)

has a unique solution that is obtained by

$$begin{aligned} begin{aligned} u(kappa )={}&{-}frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+frac{delta }{gamma } sum _{xi =0}^{ell +1} mathcal{F}(xi +vartheta -1) \ &{}+frac{1}{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1). end{aligned} end{aligned}$$

(5)

Proof

Let (u(kappa )) be a solution to (4). Using Lemma 2.4, for some constants (mathcal{A}_{i} in mathbb{R}), for (i=0, 1), we have

$$begin{aligned} u(kappa )=-Delta ^{-vartheta } mathcal{F}(kappa +vartheta -1)+ mathcal{A}_{0}+mathcal{A}_{1} kappa. end{aligned}$$

Using the fractional sum of order (vartheta in (1,2]), we obtain

$$begin{aligned} u(kappa )= -frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi + vartheta -1)+ mathcal{A}_{0}+mathcal{A}_{1} kappa, quadkappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}. end{aligned}$$

(6)

By applying Δ to the parts of (6), we have

$$begin{aligned} Delta u(kappa )={}&{ -}Delta ^{-(vartheta -1)}mathcal{F}(kappa + vartheta -1)+mathcal{A}_{0}Delta (1)+ mathcal{A}_{1} Delta kappa \ ={}&{-}frac{1}{Gamma (vartheta -1)}sum_{xi =0}^{kappa -vartheta +1} bigl( kappa -sigma (xi )bigr)^{(vartheta -2)}mathcal{F}(xi +vartheta -1)+ mathcal{A}_{1}. end{aligned}$$

(7)

Due to the first boundary condition (Delta u(vartheta -2)=0) in (7), we obtain (mathcal{A}_{1}=0). Using the CFDO ({}^{C}Delta ^{vartheta -1}) of order (vartheta -1in (0,1]) on both the sides of (6) with (mathcal{A}_{1}=0), it provides

$$begin{aligned} {}^{C}Delta ^{vartheta -1} u(kappa )= – {}^{C}Delta ^{vartheta -1} bigl[Delta ^{-vartheta }mathcal{F}(kappa +vartheta -1) bigr]+{}^{C} Delta ^{vartheta -1} mathcal{A}_{0}. end{aligned}$$

Here, using the Definition 2.2 that for constant (mathcal{A}_{0}), ({}^{C}Delta ^{vartheta -1}mathcal{A}_{0}= Delta ^{-(2-vartheta )} Delta mathcal{A}_{0}=Delta ^{-(2-vartheta )}(0) = 0), yields

$$begin{aligned} {}^{C} Delta ^{vartheta -1} u(kappa )= -Delta ^{-1}mathcal{F}( kappa +vartheta -1)=-sum _{xi =0}^{kappa -1}mathcal{F}(xi + vartheta -1). end{aligned}$$

(8)

Using the second boundary condition (delta {}^{C}Delta ^{ vartheta -1} u(vartheta +ell )+gamma u(eta )=0) in (6) and (8), we obtain

$$begin{aligned} delta {}^{C}Delta ^{vartheta -1} u(vartheta +ell )= -delta sum _{xi =0}^{vartheta +ell -1}mathcal{F}(xi +vartheta -1)= – delta sum_{xi =0}^{ lceil (vartheta -1)+ell rceil } mathcal{F}(xi + vartheta -1). end{aligned}$$

Since (vartheta -1leq 1), we obtain

$$begin{aligned} delta {}^{C}Delta ^{vartheta -1} u(vartheta +ell )= -delta sum_{xi =0}^{ell +1}mathcal{F}(xi + vartheta -1) end{aligned}$$

(9)

and

$$begin{aligned} gamma u(eta )= -frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{ eta -vartheta }bigl(eta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+gamma mathcal{A}_{0}. end{aligned}$$

(10)

From (9) and (10) in (delta {}^{C}Delta ^{vartheta -1} u(vartheta +ell )+u(eta )=0), we arrive at

$$begin{aligned} -delta sum_{xi =0}^{ell +1}mathcal{F}(xi + vartheta -1)- frac{gamma }{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta } bigl( eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1)+ gamma mathcal{A}_{0}=0. end{aligned}$$

This leads to

$$begin{aligned} mathcal{A}_{0}=frac{delta }{gamma } sum _{xi =0}^{ell +1} mathcal{F}(xi +vartheta -1)+ frac{1}{Gamma (vartheta )}sum_{ xi =0}^{eta -vartheta }bigl(eta -sigma (xi )bigr)^{(vartheta -1)} mathcal{F}(xi +vartheta -1). end{aligned}$$

(11)

Using the values of (mathcal{A}_{i} in mathbb{R}), for (i=0, 1) in (u(kappa )), we obtain

$$begin{aligned} begin{aligned} u(kappa )={}&{-}frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+frac{delta }{gamma } sum _{xi =0}^{ell +1} mathcal{F}(xi +vartheta -1) \ &{}+frac{1}{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1), end{aligned} end{aligned}$$

(12)

for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}). The proof is completed. □

We introduce the notation (Phi _{u}^{vartheta }(kappa )=mathcal{F}(kappa +vartheta -1, u( kappa +vartheta -1))). To transform the above DFTM with three-point BCs (2) to a fixed-point theorem, we define the operator (mathcal{T}:mathcal{B}rightarrow mathcal{B}) by

$$begin{aligned} begin{aligned} (mathcal{T}u) (kappa )={}&{-} frac{1}{Gamma (vartheta )} sum_{xi =0}^{kappa -vartheta }bigl( kappa -sigma (xi )bigr)^{( vartheta -1)}Phi _{u}^{vartheta }(xi )+frac{delta }{gamma } sum_{xi =0}^{ell +1}Phi _{u}^{vartheta }(xi ) \ &{}+ frac{1}{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)}Phi _{u}^{vartheta }(xi ), end{aligned} end{aligned}$$

(13)

for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}). We know that the fixed point of (mathcal{T}) is a solution to (2).

We consider the following hypotheses:

((mathcal{H}_{1})):

The Lipschitz-type inequality: There exists (mathcal{K}>0) such that (vert mathcal{F}(kappa,u)-mathcal{F}(kappa,hat{u}) vert leq mathcal{K} vert u-hat{u} vert ) for all (u,hat{u}in mathcal{B}) and each (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}).

((mathcal{H}_{2})):

There exists a bounded function (mathcal{L}:mathbb{N}_{vartheta -2}^{vartheta +ell +1} rightarrow mathbb{R}) with (vert mathcal{F}(kappa,u) vert leq mathcal{L}(kappa ) vert u vert ) for all (uin mathcal{B}).

Theorem 3.2

If the hypothesis ((mathcal{H}_{1})) holds, then the DFTM with threepoint BCs (2) has a unique solution in (mathcal{B}) provided

$$begin{aligned} begin{aligned} frac{1}{Gamma (vartheta +1)} bigl[(vartheta + ell +1)^{( vartheta )}+eta ^{(vartheta )} bigr]+frac{delta }{gamma } ( ell +2)< frac{1}{mathcal{K}}. end{aligned} end{aligned}$$

(14)

Proof

Let (u,hat{u}in mathcal{B}). Then, for each (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}), we have

$$begin{aligned} begin{aligned} biglvert (mathcal{T}u) (kappa )-( mathcal{T}hat{u}) ( kappa ) bigrvert leq {}& frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)} biglvert Phi _{u}^{vartheta }(xi )-Phi _{hat{u}}^{vartheta }(xi ) bigrvert \ &{}+frac{delta }{gamma } sum_{xi =0}^{ell +1}(1) biglvert Phi _{u}^{ vartheta }(xi )-Phi _{hat{u}}^{vartheta }( xi ) bigrvert \ &{}+frac{1}{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)} biglvert Phi _{u}^{vartheta }( xi )-Phi _{hat{u}}^{vartheta }(xi ) bigrvert , end{aligned} end{aligned}$$

(15)

where (Phi _{u}^{vartheta },Phi _{hat{u}}^{vartheta }in mathcal{C} (mathbb{N}_{vartheta -2}^{vartheta +ell +1}, mathbb{R} )) satisfies the functional equations

$$begin{aligned} Phi _{u}^{vartheta }(kappa )=mathcal{F}bigl( kappa +vartheta -1, u( kappa +vartheta -1)bigr),qquad Phi _{hat{u}}^{vartheta }( kappa )= mathcal{F}bigl(kappa +vartheta -1, hat{u}(kappa +vartheta -1) bigr). end{aligned}$$

(16)

From the assumption ((mathcal{H}_{1})), we obtain

$$begin{aligned} biglvert Phi _{u}^{vartheta }(kappa )-Phi _{hat{u}}^{vartheta }( kappa ) bigrvert &= biglvert mathcal{F}bigl(kappa +vartheta -1, u(kappa + vartheta -1)bigr)-mathcal{F} bigl(kappa +vartheta -1, hat{u}(kappa + vartheta -1)bigr) bigrvert \ &leq mathcal{K} biglvert u(kappa +vartheta -1)-hat{u}(kappa + vartheta -1) bigrvert , \ biglvert Phi _{u}^{vartheta }(kappa )-Phi _{hat{u}}^{vartheta }( kappa ) bigrvert &leq mathcal{K} Vert u-hat{u} Vert . end{aligned}$$

(17)

Substituting the inequality (17) into (15), it follows that

$$begin{aligned} begin{aligned} Vert mathcal{T}u-mathcal{T}hat{u} Vert leq {}& frac{mathcal{K} Vert u-hat{u} Vert }{Gamma (vartheta )} sum_{xi =0}^{kappa -vartheta }bigl( kappa -sigma (xi )bigr)^{( vartheta -1)}+frac{delta }{gamma } mathcal{K} Vert u-hat{u} Vert sum_{xi =0}^{ell +1}(1) \ &{}+frac{mathcal{K} Vert u-hat{u} Vert }{Gamma (vartheta )} sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)}. end{aligned} end{aligned}$$

In view of Lemma 2.5 of (a), we obtain

$$begin{aligned} begin{aligned} Vert mathcal{T}u-mathcal{T}hat{u} Vert &leq biggl[frac{1}{Gamma (vartheta +1)} bigl(kappa ^{(vartheta )}+ eta ^{(vartheta )} bigr)+frac{delta }{gamma } (ell +2) biggr] mathcal{K} Vert u-hat{u} Vert \ &leq biggl[ frac{1}{Gamma (vartheta +1)} bigl[(vartheta + ell +1)^{(vartheta )}+eta ^{(vartheta )} bigr]+ frac{delta }{gamma } (ell +2) biggr]mathcal{K} Vert u-hat{u} Vert , end{aligned} end{aligned}$$

therefore, it follows that (mathcal{T}) is a contraction and has a unique fixed point that is the solution of (2). □

Theorem 3.3

The DFTM with threepoint BCs (2) has at least one solution under the assumption ((mathcal{H}_{2})) and the inequality

$$begin{aligned} begin{aligned} mathcal{L}^{*} leq frac{gamma Gamma (vartheta +1)}{gamma [ (vartheta +ell +1)^{(vartheta )}+eta ^{(vartheta )} ]+delta (ell +2)Gamma (vartheta +1)}, end{aligned} end{aligned}$$

(18)

where (mathcal{L}^{*}=max { mathcal{L}(kappa ):mathbb{N}_{ vartheta -2}^{vartheta +ell +1} } ).

Proof

Suppose that (mathfrak{M}>0) and (mathcal{S}_{u}= { u(kappa ) | mathbb{N}_{vartheta -2}^{ vartheta +ell +1}rightarrow mathbb{R}, Vert u Vert leq mathfrak{M} } ). We must first show that (mathcal{T}) maps (mathcal{S}_{u}) in (mathcal{S}_{u}).

For (u(kappa )in mathcal{S}_{u}), we have

$$begin{aligned} begin{aligned} biglvert (mathcal{T}u) (kappa ) bigrvert leq {}& frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa -vartheta } bigl( kappa -sigma (xi )bigr)^{(vartheta -1)} biglvert Phi _{u}^{vartheta }( xi ) bigrvert +frac{delta }{gamma } sum _{xi =0}^{ell +1}(1) biglvert Phi _{u}^{vartheta }(xi ) bigrvert \ &{}+frac{1}{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)} biglvert Phi _{u}^{vartheta }( xi ) bigrvert , end{aligned} end{aligned}$$

(19)

where (Phi _{u}^{vartheta }(kappa )) is given in (16). Using ((mathcal{H}_{2})), we arrive at

$$begin{aligned} begin{aligned} biglvert Phi _{u}^{vartheta }( kappa ) bigrvert = biglvert mathcal{F}bigl(kappa +vartheta -1, u( kappa +vartheta -1)bigr) bigrvert leq mathcal{L}(kappa ) biglvert u( kappa +vartheta -1) bigrvert leq mathcal{L}^{*} Vert u Vert . end{aligned} end{aligned}$$

(20)

Hence, putting the inequality (19) and (20) together, we conclude that

$$begin{aligned} Vert mathcal{T}u Vert leq Biggl[ frac{1}{Gamma (vartheta )} Biggl(sum _{xi =0}^{kappa – vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}+sum_{xi =0}^{ eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)} Biggr)+ frac{delta }{gamma } sum _{xi =0}^{ell +1}(1) Biggr]mathcal{L}^{*} Vert u Vert . end{aligned}$$

From Lemma 2.5 of (a), we have

$$begin{aligned} begin{aligned} Vert mathcal{T}u Vert &leq biggl[ frac{1}{Gamma (vartheta +1)} bigl(kappa ^{(vartheta )}+eta ^{( vartheta )} bigr)+ frac{delta }{gamma } (ell +2) biggr] mathcal{L}^{*} Vert u Vert \ &leq frac{gamma [(vartheta +ell +1)^{(vartheta )}+eta ^{(vartheta )} ]+delta (ell +2)Gamma (vartheta +1)}{gamma Gamma (vartheta +1)} mathcal{L}^{*}mathfrak{M}. end{aligned} end{aligned}$$

In view of (18), we obtained (Vert mathcal{T}u Vert leq mathfrak{M}). Thus, (mathcal{T}) maps (mathcal{S}_{u}) in (mathcal{S}_{u}) and has at least one fixed point that is a solution to (2), according to the Brouwer fixed-point theorem. □

Thermostat model with two sensors

This section discusses the stability results for the DFTM with four-point BCs (3).

Theorem 3.4

Assume (mathcal{F}: mathbb{N}_{vartheta -2}^{vartheta +ell +1} rightarrow mathbb{R}) is given. A unique solution to the DFTM with fourpoint BCs

$$begin{aligned} textstylebegin{cases} {}^{C}Delta ^{vartheta }u(kappa )=mathcal{F}(kappa +vartheta -1),quad kappa in {mathbb{N}_{0}^{ell +1}}, \ Delta u(vartheta -2)=beta u(zeta ),qquad delta {}^{C}Delta ^{ vartheta -1} u(vartheta +ell )+gamma u(eta )=0, end{cases}displaystyle end{aligned}$$

(21)

has the form

$$begin{aligned} begin{aligned} u(kappa )={}& Biggl[frac{1}{Gamma (vartheta )} sum_{ xi =0}^{kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}+ frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )}sum_{ xi =0}^{zeta -vartheta } bigl(zeta -sigma (xi )bigr)^{(vartheta -1)} Biggr]mathcal{F}(xi +vartheta -1) \ &{}-mathcal{D}_{2}(kappa ) Biggl[frac{gamma }{Gamma (vartheta )} sum _{xi =0}^{eta -vartheta }bigl(eta -sigma (xi ) bigr)^{(vartheta -1)}+ delta sum_{xi =0}^{ell +1}(1) Biggr]mathcal{F}(xi +vartheta -1), end{aligned} end{aligned}$$

(22)

where (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}), (mathcal{D}_{1}(kappa )= frac{[delta mu +gamma (eta -kappa )]}{mathcal{Q}}), (mathcal{D}_{2}(kappa )= frac{[beta (zeta -kappa )-1]}{mathcal{Q}}) such that (mathcal{Q}=gamma (beta zeta -1)-beta ( delta mu + gamma eta )) and (mu =frac{1}{Gamma (3-vartheta )}(vartheta +ell )^{(2- vartheta )}).

Proof

For the fractional sum of order (vartheta in (1,2]) for (21) and using Lemma 2.4, we obtain

$$begin{aligned} u(kappa )= frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi + vartheta -1)+ mathcal{A}_{2}+mathcal{A}_{3} kappa, end{aligned}$$

(23)

where (mathcal{A}_{i} in mathbb{R}), for (i=2,3). Applying the operators Δ and ({}^{C}Delta ^{vartheta -1}) on both sides of (23) together with Definitions 2.1 and 2.2, we obtain

$$begin{aligned} Delta u(kappa )=frac{1}{Gamma (vartheta -1)}sum _{xi =0}^{ kappa -vartheta +1}bigl(kappa -sigma (xi ) bigr)^{(vartheta -2)} mathcal{F}(xi +vartheta -1)+ mathcal{A}_{3} end{aligned}$$

(24)

and

$$begin{aligned} {}^{C}Delta ^{vartheta -1} u(kappa )= sum _{xi =0}^{kappa -1} mathcal{F}(xi +vartheta -1)+ mathcal{A}_{3} mu. end{aligned}$$

(25)

In view of (Delta u(vartheta -2)= beta u(zeta )), we obtain

$$begin{aligned} beta u(zeta )= frac{beta }{Gamma (vartheta )}sum _{xi =0}^{ zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+beta mathcal{A}_{2}+ beta mathcal{A}_{3} zeta end{aligned}$$

(26)

and

$$begin{aligned} Delta u(vartheta -2)= mathcal{A}_{3}. end{aligned}$$

(27)

From (26) and (27) and employing the first boundary condition (21), we obtain

$$begin{aligned} frac{beta }{Gamma (vartheta )}sum_{xi =0}^{zeta -vartheta } bigl( zeta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1)+ beta mathcal{A}_{2}+mathcal{A}_{3}(beta zeta -1)=0. end{aligned}$$

(28)

In view of (delta {}^{C}Delta ^{vartheta -1} u(vartheta +ell )+gamma u( eta )=0), we obtain

$$begin{aligned} gamma u(eta )= frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{ eta -vartheta }bigl(eta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+gamma mathcal{A}_{2}+ gamma mathcal{A}_{3} eta end{aligned}$$

(29)

and

$$begin{aligned} delta {}^{C} Delta ^{vartheta -1} u(vartheta +ell ) = delta sum _{xi =0}^{ lceil (vartheta -1)+ell rceil } mathcal{F}(xi +vartheta -1)+ delta mathcal{A}_{3} mu. end{aligned}$$

Since (vartheta -1leq 1), we arrive at

$$begin{aligned} delta Delta ^{vartheta -1} u(vartheta +ell )= delta sum _{xi =0}^{ ell +1}mathcal{F}(xi +vartheta -1)+ delta mathcal{A}_{3} mu. end{aligned}$$

(30)

From (29) and (30) with the help of the second boundary condition (21), we have

$$begin{aligned} &gamma mathcal{A}_{2}+mathcal{A}_{3}( delta mu +gamma eta ) \ &quad{} +frac{gamma }{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta } bigl( eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1)+ delta sum_{xi =0}^{ell +1}mathcal{F}(xi + vartheta -1)=0. end{aligned}$$

(31)

The constant (mathcal{A}_{3}) can be obtained by solving equations (28) and (31),

$$begin{aligned} &mathcal{A}_{3} mathcal{Q}+ frac{gamma beta }{Gamma (vartheta )}sum _{xi =0}^{zeta – vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}(xi + vartheta -1) \ &quad{} -frac{gamma beta }{Gamma (vartheta )}sum_{xi =0}^{eta – vartheta } bigl(eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi + vartheta -1)- delta beta sum_{xi =0}^{ell +1}mathcal{F}(xi + vartheta -1)=0, end{aligned}$$

which implies

$$begin{aligned} &mathcal{A}_{3}=frac{1}{mathcal{Q}} Biggl[ frac{gamma beta }{Gamma (vartheta )}sum_{xi =0}^{eta – vartheta }bigl(eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}(xi + vartheta -1) \ &quad{}times delta beta sum_{xi =0}^{ell +1} mathcal{F}(xi + vartheta -1) -frac{gamma beta }{Gamma (vartheta )}sum _{xi =0}^{ zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1) Biggr]. end{aligned}$$

(32)

Substituting (mathcal{A}_{3}) into (28), we have

$$begin{aligned} beta mathcal{A}_{2}={}& frac{beta [gamma (beta zeta -1)-mathcal{Q}]}{mathcal{Q} Gamma (vartheta )} sum _{xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} mathcal{F}(xi +vartheta -1) \ &{} -frac{beta [beta zeta -1]}{mathcal{Q}} Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}mathcal{F}(xi +vartheta -1)+ delta sum _{xi =0}^{ell +1}mathcal{F}(xi +vartheta -1) Biggr]. end{aligned}$$

This implies,

$$begin{aligned} mathcal{A}_{2}={}&frac{1}{mathcal{Q}} Biggl[ frac{beta (delta mu +gamma eta )]}{ Gamma (vartheta )}sum _{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} mathcal{F}(xi +vartheta -1)- [beta zeta -1] \ &{} times Biggl(frac{gamma }{Gamma (vartheta )}sum_{xi =0}^{ eta -vartheta } bigl(eta -sigma (xi )bigr)^{(vartheta -1)}mathcal{F}( xi +vartheta -1)+ delta sum_{xi =0}^{ell +1}mathcal{F}(xi + vartheta -1) Biggr) Biggr]. end{aligned}$$

Using the constants (mathcal{A}_{i} in mathbb{R}), for (i=2, 3) in (23), we obtain u in the form

$$begin{aligned} begin{aligned} u(kappa )={}& Biggl[frac{1}{Gamma (vartheta )}sum _{ xi =0}^{kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}+ frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )}sum_{ xi =0}^{zeta -vartheta } bigl(zeta -sigma (xi )bigr)^{(vartheta -1)} Biggr]mathcal{F}(xi +vartheta -1) \ &{}-mathcal{D}_{2}(kappa ) Biggl[frac{gamma }{Gamma (vartheta )} sum _{xi =0}^{eta -vartheta }bigl(eta -sigma (xi ) bigr)^{(vartheta -1)}+ delta sum_{xi =0}^{ell +1}(1) Biggr]mathcal{F}(xi +vartheta -1), end{aligned} end{aligned}$$

for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}). □

We assume that (mathcal{F}) is a real-valued continuous function on ({mathbb{N}_{vartheta -2}^{vartheta +ell +1}}) such that (Phi _{hat{u}}^{vartheta }(kappa )=mathcal{F}(kappa +vartheta -1, hat{u}(kappa +vartheta -1))). Now, we introduce the definitions of Ulam stability for DFC given on the basis of [60, 61].

Definition 3.5

If for every function (hat{u}(kappa )in mathbb{B}) of

$$begin{aligned} biglvert {}^{C}Delta ^{vartheta } hat{u}(kappa )-Phi _{hat{u}}^{ vartheta }(xi ) bigrvert leq epsilon, end{aligned}$$

(33)

where (kappa in {mathbb{N}_{0}^{ell +1}}), (epsilon >0), there exists a solution (u(kappa )in mathbb{B}) of (3) and a positive constant (mathcal{P}_{1}>0) such that

$$begin{aligned} biglvert hat{u}(kappa )-u(kappa ) bigrvert leq mathcal{P}_{1} epsilon,quad kappa in {mathbb{N}_{vartheta -2}^{vartheta +ell +1}}. end{aligned}$$

(34)

Then, the DFTM with four-point BCs (3) is Hyers–Ulam (HU) stable. Equation (3) is also said to be generalized HU stable if we substitute (Theta (epsilon )=mathcal{P}_{1} epsilon ) in inequality (34), where (Theta (epsilon )in mathbb{C} (mathbb{R}^{+}, mathbb{R}^{+} )) and (Theta (0)=0).

Definition 3.6

Let (hat{u}(kappa )in mathbb{B}), then the following inequality holds

$$begin{aligned} biglvert {}^{C}Delta ^{vartheta } hat{u}(kappa )-Phi _{hat{u}}^{ vartheta }(xi ) bigrvert leq epsilon phi ( kappa +vartheta -1), end{aligned}$$

(35)

where (kappa in {mathbb{N}_{0}^{ell +1}}), (epsilon >0), there is a solution (u(kappa )in mathbb{B}) of (3) and a positive constant (mathcal{P}_{2}>0) such that

$$begin{aligned} biglvert hat{u}(kappa )-u(kappa ) bigrvert leq delta _{2} epsilon Phi (kappa +vartheta -1), quadkappa in { mathbb{N}_{vartheta -2}^{ vartheta +ell +1}}. end{aligned}$$

(36)

Then, the DFTM with four-point BCs (3) is Hyers–Ulam–Rassias (HUR) stable. Equation (3) is generalized HUR stable if we substitute (phi (kappa +vartheta -1)= epsilon phi (kappa +vartheta -1)) in inequalities (35) and (36).

Remark 3.7

A function (hat{u}(kappa )in mathcal{B}) is a solution to the inequalities (33) and (35) if there exists a function (f:mathbb{N}_{beta -2}^{beta +ell +1}rightarrow mathbb{R}) satisfying, for (kappa in mathbb{N}_{0}^{ell +1})

  1. (i)

    (vert f(kappa +vartheta -1) vert leq epsilon ),

  2. (ii)

    ({}^{C}Delta ^{vartheta } hat{u}(kappa )=Phi _{hat{u}}^{vartheta }( kappa )+f(kappa +vartheta -1)),

  3. (iii)

    (vert f(kappa +beta -1) vert leq epsilon phi (kappa +beta -1)),

  4. (iv)

    ({}^{C} Delta ^{beta } hat{u}(kappa )=Phi _{hat{u}}^{vartheta }( kappa )+f(kappa +beta -1)).

Lemma 3.8

If (hat{u}(kappa )) solves the inequality (33) for (kappa in mathbb{N}_{0}^{ell +1}), then

$$begin{aligned} begin{aligned} & Bigglvert hat{u}(kappa ) – frac{1}{Gamma (vartheta )} sum_{xi =0}^{kappa -vartheta }bigl( kappa -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }( xi )-frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )} sum_{xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi ) \ &quad{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{hat{u}}^{vartheta }( xi ) Biggr] Biggrvert leq frac{epsilon }{Gamma (vartheta +1)}( vartheta +ell +1)^{(vartheta )}, end{aligned} end{aligned}$$

where (mathcal{D}_{1}(kappa )) and (mathcal{D}_{2}(kappa )) are defined in Theorem 3.4.

Proof

If (hat{u}(kappa )) solves the inequality (33), then from (ii) of Remark 3.7 and Lemma 2.4, the solution to (ii) of Remark 3.7 satisfies

$$begin{aligned} begin{aligned} hat{u}(kappa )={}& frac{1}{Gamma (vartheta )} sum_{ xi =0}^{kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi )+ frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )}sum_{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi )bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi ) \ &{}-mathcal{D}_{2}(kappa ) Biggl[frac{gamma }{Gamma (vartheta )} sum _{xi =0}^{eta -vartheta }bigl(eta -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi )+delta sum _{xi =0}^{ell +1} Phi _{hat{u}}^{vartheta }( xi ) Biggr] \ &{}+frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa -vartheta }bigl( kappa -sigma (xi )bigr)^{(vartheta -1)}f(xi +vartheta -1). end{aligned} end{aligned}$$

(37)

Using (a) of Lemma 2.5 together with (i) of Remark 3.7, we arrive at

$$begin{aligned} & Bigglvert hat{u}(kappa ) – frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}Phi _{ hat{u}}^{vartheta }(xi )- frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )} sum_{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi ) \ &qquad{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{hat{u}}^{vartheta }( xi ) Biggr] Biggrvert \ &quad = Bigglvert frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}f(xi +vartheta -1) Biggrvert \ &quad leq frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa -vartheta } bigl( kappa -sigma (xi )bigr)^{(vartheta -1)} biglvert f(xi +vartheta -1) bigrvert \ &quad leq frac{epsilon }{Gamma (vartheta +1)}kappa ^{(vartheta )} \ &quad leq frac{epsilon }{Gamma (vartheta +1)}(vartheta +ell +1)^{( vartheta )}. end{aligned}$$

This completes the proof. □

Theorem 3.9

Assume that the following inequalities and ((mathcal{H}_{1})) hold at the same time

$$begin{aligned} begin{aligned} Lambda =mathcal{K} biggl( frac{(vartheta +ell +1)^{(vartheta )}+beta mathcal{G}_{1} zeta ^{(vartheta )}+mathcal{G}_{2} [gamma eta ^{(vartheta )}+delta (ell +2)Gamma (vartheta +1) ]}{Gamma (vartheta +1)} biggr)< 1, end{aligned} end{aligned}$$

(38)

then the DFTM with fourpoint BCs (3) is HU stable and generalized HUR stable.

Proof

From solution (22), for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}), it follows that

$$begin{aligned} biglvert hat{u}(kappa ) – u(kappa ) bigrvert leq{} & Bigglvert hat{u}( kappa ) – Biggl[frac{1}{Gamma (vartheta )}sum_{xi =0}^{ kappa -vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}+ frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )}sum _{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} Biggr]Phi _{hat{u}}^{vartheta }(xi ) \ &{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{hat{u}}^{vartheta }( xi ) Biggr] Biggrvert \ &{}+ frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa -vartheta } bigl( kappa -sigma (xi )bigr)^{(vartheta -1)} biglvert Phi _{hat{u}}^{ vartheta }(xi ) – Phi _{u}^{vartheta }(xi ) bigrvert \ &{}+ frac{beta vert mathcal{D}_{1}(kappa ) vert }{Gamma (vartheta )} sum_{xi =0}^{zeta -vartheta } bigl(zeta -sigma (xi )bigr)^{(vartheta -1)} biglvert Phi _{hat{u}}^{vartheta }(xi ) – Phi _{u}^{vartheta }(xi ) bigrvert \ &{}+ biglvert mathcal{D}_{2}(kappa ) bigrvert Biggl[ frac{gamma }{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta }bigl( eta -sigma (xi )bigr)^{(vartheta -1)}+delta sum_{xi =0}^{ell +1}(1) Biggr] biglvert Phi _{hat{u}}^{vartheta }(xi ) – Phi _{u}^{ vartheta }(xi ) bigrvert , end{aligned}$$

where (mathcal{D}_{1}(kappa )), (mathcal{D}_{2}(kappa )) are defined in Theorem 3.4 and (Phi _{u}^{vartheta }(kappa )), (Phi _{hat{u}}^{vartheta }(kappa )) are given in (16). Using the inequality (17) and Lemma 3.8 along with an application of Lemma 2.5 of (a), implies that

$$begin{aligned} Vert hat{u} – u Vert leq {}& frac{epsilon }{Gamma (vartheta +1)}(vartheta +ell +1)^{( vartheta )}+ frac{mathcal{K} Vert hat{u}-u Vert }{Gamma (vartheta +1)} bigl[(vartheta +ell +1)^{(vartheta )}+beta mathcal{G}_{1} zeta ^{(vartheta )} bigr] \ &{}+mathcal{K}mathcal{G}_{2} Vert hat{u}-u Vert biggl[ frac{gamma eta ^{(vartheta )}}{Gamma (vartheta +1)}+delta ( ell +2) biggr], end{aligned}$$

(39)

where (mathcal{G}_{1}= vert frac{delta mu +gamma [eta -(vartheta +ell +1)]}{mathcal{Q}} vert ) and (mathcal{G}_{2}= vert frac{[beta (zeta -[vartheta +ell +1])-1]}{mathcal{Q}} vert ).

Inequality (39) yields (Vert hat{u}-u Vert leq mathcal{P}_{1}epsilon ), where

$$mathcal{P}_{1}= frac{(vartheta +ell +1)^{(vartheta )}}{Gamma (vartheta +1)-mathcal{K} [(vartheta +ell +1)^{(vartheta )}+beta mathcal{G}_{1} zeta ^{(vartheta )} +mathcal{G}_{2} (gamma eta ^{(vartheta )}+delta (ell +2)Gamma (vartheta +1) ) ]}. $$

Thus, the solution to (3) is HU stable.

Further, by taking (Theta (epsilon )=mathcal{P}_{1} epsilon ) with (Theta (0)=0), we have

$$begin{aligned} Vert hat{u}- u Vert leq Theta (epsilon ). end{aligned}$$

Hence, the solution to (3) becomes generalized HU stable. □

Finally, we consider the following hypotheses to discuss the HUR stability and generalized HUR stability in the next results.

((mathcal{H}_{3})):

For an increasing function (phi in mathcal{C} ({mathbb{N}_{vartheta -2}^{vartheta + ell }}, mathbb{R}^{+} )), there exists (lambda _{phi }>0) such that, for (kappa in {mathbb{N}_{0}^{ell +1}})

  1. (i)

    (frac{epsilon }{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta }(kappa -sigma (xi ))^{(vartheta -1)}phi (xi + vartheta -1)leq lambda _{phi } epsilon phi (kappa +vartheta -1)), consequently

  2. (ii)

    (frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta }(kappa -sigma (xi ))^{(vartheta -1)}phi (xi + vartheta -1)leq lambda _{phi } phi (kappa +vartheta -1)).

Lemma 3.10

If (hat{u}(kappa )) solves the inequality (35) for (kappa in mathbb{N}_{0}^{ell +1}), then

$$begin{aligned} & Bigglvert hat{u}(kappa ) – frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}Phi _{ hat{u}}^{vartheta }(xi )- frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )} sum_{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi ) \ &quad{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{hat{u}}^{vartheta }( xi ) Biggr] Biggrvert leq lambda _{phi }epsilon phi (kappa + vartheta -1), end{aligned}$$

where (mathcal{D}_{1}(kappa )) and (mathcal{D}_{2}(kappa )) are defined in Theorem 3.4.

Proof

From inequality (35), we obtain a solution to (iv) of Remark 3.7 that satisfies (37). Using ((mathcal{H}_{3})) of (i), for (kappa in mathbb{N}_{0}^{ell +1}) and Remark 3.7 of (iii), it follows that

$$begin{aligned} &Bigglvert hat{u}(kappa ) – frac{1}{Gamma (vartheta )}sum _{xi =0}^{ kappa -vartheta }bigl(kappa -sigma (xi ) bigr)^{(vartheta -1)}Phi _{ hat{u}}^{vartheta }(xi )- frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )} sum_{ xi =0}^{zeta -vartheta }bigl(zeta -sigma (xi ) bigr)^{(vartheta -1)} Phi _{hat{u}}^{vartheta }(xi ) \ &qquad{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{hat{u}}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{hat{u}}^{vartheta }( xi ) Biggr] Biggrvert \ &quad leq frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa -vartheta } bigl( kappa -sigma (xi )bigr)^{(vartheta -1)} biglvert f(xi +vartheta -1) bigrvert \ &quad leq frac{epsilon }{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}phi (xi + vartheta -1) \ &quadleq lambda _{phi }epsilon phi (kappa +vartheta -1). end{aligned}$$

This completes the proof. □

Theorem 3.11

If the hypothesis ((mathcal{H}_{1})) holds with the inequality (38), then the DFTM with fourpoint BCs (3) is HUR stable and generalized HUR stable.

Proof

From the solution (22), for (kappa in mathbb{N}_{vartheta -2}^{vartheta +ell +1}), we obtain

$$begin{aligned} biglvert hat{u}(kappa ) – u(kappa ) bigrvert leq {}& Bigglvert hat{u}( kappa ) – frac{1}{Gamma (vartheta )}sum_{xi =0}^{kappa – vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)}Phi _{u}^{ vartheta }( xi ) \ &{} -frac{beta mathcal{D}_{1}(kappa )}{Gamma (vartheta )}sum_{ xi =0}^{zeta -vartheta } bigl(zeta -sigma (xi )bigr)^{(vartheta -1)} Phi _{u}^{vartheta }( xi ) \ &{} +mathcal{D}_{2}(kappa ) Biggl[ frac{gamma }{Gamma (vartheta )}sum _{xi =0}^{eta -vartheta }bigl( eta -sigma (xi ) bigr)^{(vartheta -1)}Phi _{u}^{vartheta }(xi )+ delta sum _{xi =0}^{ell +1}Phi _{u}^{vartheta }( xi ) Biggr] Biggrvert , end{aligned}$$

where (mathcal{D}_{1}(kappa )) and (mathcal{D}_{2}(kappa )) are defined in Theorem 3.4. Using Lemma 3.10 and the procedure used in Theorem 3.9, we obtain

$$begin{aligned} Vert hat{u} – u Vert leq {}& lambda _{phi }epsilon phi ( kappa +vartheta -1)+frac{mathcal{K}}{Gamma (vartheta )}sum_{ xi =0}^{kappa -vartheta } bigl(kappa -sigma (xi )bigr)^{(vartheta -1)} Vert hat{u} – u Vert \ &{}+ frac{mathcal{K}beta vert mathcal{D}_{1}(kappa ) vert }{Gamma (vartheta )} sum_{xi =0}^{zeta -vartheta } bigl(zeta -sigma (xi )bigr)^{( vartheta -1)} Vert hat{u} – u Vert \ &{}+mathcal{K} biglvert mathcal{D}_{2}(kappa ) bigrvert Biggl[ frac{gamma }{Gamma (vartheta )}sum_{xi =0}^{eta -vartheta } bigl( eta -sigma (xi )bigr)^{(vartheta -1)}+delta sum _{xi =0}^{ell +1}(1) Biggr] Vert hat{u} – u Vert . end{aligned}$$

By an application of Lemma 2.5 of (a), the above inequality becomes

$$begin{aligned} begin{aligned} Vert hat{u} – u Vert leq{} & lambda _{phi }epsilon phi (kappa +vartheta -1)+ frac{mathcal{K} Vert hat{u}-u Vert }{Gamma (vartheta +1)} bigl[(vartheta +ell +1)^{(vartheta )}+beta mathcal{G}_{1} zeta ^{(vartheta )} bigr] \ &{}+mathcal{K}mathcal{G}_{2} Vert hat{u}-u Vert biggl[gamma frac{eta ^{(vartheta )}}{Gamma (vartheta +1)}+delta (ell +2) biggr], end{aligned} end{aligned}$$

(40)

where (mathcal{G}_{1}) and (mathcal{G}_{2}) are defined in Theorem 3.9. From which, the inequality (40) yields

$$begin{aligned} Vert hat{u}- u Vert leq mathcal{P}_{2} epsilon phi ( kappa +vartheta -1), end{aligned}$$

where (mathcal{P}_{2} = frac{lambda _{phi }Gamma (vartheta +1)}{Gamma (vartheta +1)-mathcal{K} [(vartheta +ell +1)^{(vartheta )}+beta mathcal{G}_{1} zeta ^{(vartheta )}+mathcal{G}_{2} (gamma eta ^{(vartheta )}+delta (ell +2)Gamma (vartheta +1) ) ]}).

Hence, the solution of (3) is HUR stable.

Also, by setting (phi (kappa +vartheta -1)=epsilon phi (kappa +vartheta -1)), we have

$$begin{aligned} Vert hat{u} – u Vert leq mathcal{P}_{2} phi (kappa + vartheta -1). end{aligned}$$

Therefore, the solution of (3) is generalized HUR stable. □

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